On average, $15\%$ of the patients a certain emergency room receives are uninsured. Let $N$ be the number of patients the emergency room receives until it receives the first uninsured patient of the day. Assume the insurance status of the patients is independent. Find the probability that it takes at least $6$ patients until the emergency room receives the first uninsured patient. You may round your answer to the nearest hundredth. $P(N\geq 6)=$
Without a fancy calculator For each patient: $P({\text{uninsured}})=0.15$ $P(\text{insured}})=0.85$ If it takes at least $6$ patients before the emergency room receives the first uninsured patient, then the first $5$ patients must be insured. $\begin{aligned} P(N\geq 6)&=P(5\text{ insured}) \\\\ &=(0.85})^5 \\\\ &= 0.4437053125 \end{aligned}$ $P(N\geq 6) \approx0.44$